Sphere in cone

A sphere of radius 3 cm desribe cone with minimum volume. Determine cone dimensions.

Correct result:

r =  4.24 cm
h =  12 cm

Solution:

tanϕ=r/h tanϕ/2=3/r  V=13Sh=13πr2h V=13r3tanϕ V=13(r/tanϕ2)3tanϕ  V=9πtan3ϕ2(1cos2ϕ32tanϕtan1(ϕ/2)1cos2(ϕ/2)) V=0  cos2(ϕ/2)32tanϕtan1(ϕ/2)cos2ϕ=0 cos(ϕ/2)sin(ϕ/2)32cosϕsinϕ=0  12sinϕ32cosϕsinϕ=0 13cosϕ=0  ϕ=arccos13=1.2309594173=703144"  r=3/tan(703144"/2)=4.24 cm
h=rtan703144"=12 cm



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